I am making this list as an attempt to fill the void of any such open source list. Your contribution is vital to keep the list up-to-date for future students to use. Hence it's a humble request to each user to contribute any new opportunity you discover and update the old ones. Without updation, the web-links will break, and the list will become futile! I'll make sure that any such modification in the README.md raised as an issue or submitted as a pull request be timely reviewed and added.

The following are some global research internship positions:

CANADA

• MITACS Globalink, for research positions in Canada.
• University of Alberta Research Experience (UARE)

US

• LPI Summer Intern Program in Planetary Science, for both national and international students.
• NASA - CalTech Summer Undergraduate Research Fellowship (SURF), for research positions at Jet Propulsion Laboratory, California Institute of Technology.
• Santa Fe Institute Summer Research Experience
• Robotics Institute for Summer Scholars, CMU
• Data Science for Social Good Fellowship, CMU
• QRLSSP Summer Program, Arizona State University: Only for US Citizens / Permanent Residents
• Summer Undergraduate Research fellowship program, Rockfeller Institute
• CalTech SURF Program
• Space Astronomy Summer Program, Space Telescope Science Institute (STScI)
• Data Visualization Programming Summer Student Internship in New York City, Flatiron Institute. (NEW LINK NEEDED)
• Summer Research Program, Princeton University
• International Student Research Internship Program - McKelvey School of Engineering
• Luddy School of Informatics, Computing, and Engineering - Global Talent Attraction Program (GTAP)
• BioChemCoRe (Biology and Chemistry Computational Research), Amaro Lab, UC San Diego
• Global Research Experience in Advanced Technologies (GREAT) Program, UC Davis
• ICT Summer Research Program
• UC Berkeley Amgen Scholars program
• Summer Undergraduate Program in Engineering Research at Berkeley (SUPERB) (for Information Technology students
• SUMMER RESEARCH OPPORTUNITIES PROGRAM(SROP), Purdue University
• Summer Undergraduate Research Fellowship (SURF), Stanford University
• IPAM UCLA Program

SWITZERLAND

• Research fellowship at ETH Zurich
• CERN Summer Student Program
• CERN OpenLab for Computer Science Undergraduates
• E3, EPFL (Switzerland)
• Summer at EPFL
• Swissnex Program

GERMANY

• Max Planck Institute for Gravitational Physics, for all students.
• Max Planck Institute for Software Systems
• Warwick Statistics Internship Scheme, University of Warwick
• DAAD WISE Scholarship
• HZDR Dresden, Summer student program (Germany)
• MaxSIP (Max Planck Institute Summer Internship)
• UROP International, RWTH Aachen

UK

• DSSGx UK Summer Fellowship, University of Warwick
• Visual Geometry Group

INDIA

• ICTS LONG TERM VISITING STUDENTS PROGRAM for both International and Indian Students. Aims to provide opportunities to students of science, mathematics and engineering to spend a longer period of time (1 or 2 semesters).
• ICTS SN BHATT Memorial Excellence Fellowship for Undergraduate students of science, mathematics and engineering, who are in their third, fourth or fifth year of the program are eligible to apply.
• INSA-IASc-NASI SRFP for undergraduate and postgraduate students in Chemistry, Earth and Planetary sciences, Engineering including computer sciences, Life sciences, mathematics and physics. The fellowship period is 56 days and is quite prestigious for anyone who wishes to go into the field of research.

SOUTH KOREA

• SPIKE @ UNIST, Summer Program for Internship and Korean Experience
• SPIKE@UNIST (South Korea)
• GIST Global Internship Program (South Korea)
• Research internship @ Yonsei University
• Yonsei University

TAIWAN

• NTHU Summer Internship Program
• TIGP-IIP, Academia Sincia, Taiwan
• ITRI Global Internship program
• TEEP@Asia (Taiwan)
• TEEP Internship, Taiwan

JAPAN

• University of Tokyo, Amgen Scholarship Japan (only Bio Tech and Bio-Chem Programs)
• OIST Research Internship Program, Okinawa Institute of Science and Technology
• NIMS (Japan)
• Hennge Global Internship Program, for remote internship in a Japan-based tech start-up.

AUSTRIA

• ISternship Summer Student Program, IST Austria

HONG KONG

• HKU, Computer Science Department
• IVISP, HKUST (only for senior year undergraduates + postgraduates)
• SURP, CUHK
• SURP (Hongkong)

AUSTRALIA

• UNSW Civil and Environmental Engineering Research Internship
• UNSW Research Internship (Australia)

ISRAEL

• Summer Intern at the Kupcinet Getz International Summer Science School of the Weizmann Institute of Science

SPAIN

• Summer Training Program,CNIO
• Computer Vision Center Internship Program

FRANCE

• OECD Internship Programme (France)

SAUDI ARABIA

• KAUST (Saudi Arabia)

RUSSIA

• HSE University, Moscow Computer Science Internships

MULTIPLE COUNTRIES

• IPAM UCLA, RIPS Program
• Amgen Scholars Program
• UX Research Internship, Red Hat
• Allen Institute for AI, Research and Engineering Internships

The following are some research internships for Indian students:

1. S.N. Bose Scholarship sponsored by IUSSTF, it's been closed for almost two years now.
2. ICTS Summer research program, by ICTS-TIFR.
3. DAAD Wise Scholarship, for research positions in Germany
4. Shastri Research Student Fellowship by Shastri Indo-Canadian Institute
5. Viterbi India Program, sponsored by IUSSTF and USC Viterbi
6. Khorana Program for Scholars, sponsored by IUSSTF
7. Charpak Global Scholarship, funded by French Embassy in India
8. Indian Student Internship Program at NTHU
9. IBM Blue Mix Research Internship
10. IBM Extreme Blue (limited to IITs and NITs)
11. LIGO IndiGo at CalTech, funded by US NSF
12. IIITD Summer Internship, IIIT, New Delhi
13. Summer Research Fellowship Programme 2021, IISc, IAS, INS
14. Quantitative Research Summer Internship, WorldQuant India
15. Internship At CMI, Chennai Mathematical Institute
16. Shastri Student Internship Project, Eligible universities here
17. Cisco Summer internship
18. Hackerrank SDE intern
19. SSERD Research Intern for research work in topics like Space Settlement, Astrophysics, Space Mission Design etc.
20. IISER Kolkata Summer Student Research Programme, IISER Kolkata
21. IITM Summer Fellowship Programme, IIT Madras
22. IITD GIPEDI, IIT Delhi
23. IITB Research Internship Award, IIT Bombay
24. IITK SURGE Program, IIT Kanpur
25. SPARK, IIT Roorkee
26. SRIP, IIT Gandhinagar
27. IIT Ropar Summer Internship [Information], IIT Ropar
28. Bhaba Atomic Research Centre
29. ISI Kolkata Summer Internship
30. IIT (ISM) Dhanbad SRIP Programme
31. IIIT Delhi
32. IIIT HYDERABAD
33. Summer Program, CeNSE IISc Bangalore